Mobile Icosapods

M. Galleta, G. Nawratilb, J. Schichoc, J.M. Seligd

a Johann Radon Institute for Computational and Applied Mathematics (RICAM), AustrianAcademy of Sciences, Altenberger Straße 69, 4040 Linz, AT.

b Institute of Discrete Mathematics and Geometry, Vienna University of Technology,Wiedner Hauptstraße 8-10/104, 1040 Vienna, AT.

c Research Institute for Symbolic Computation, Johannes Kepler University, AltenbergerStraße 69, 4040 Linz, AT.

d School of Engineering, London South Bank University, London SE1 0AA, U.K.

Abstract

Pods are mechanical devices constituted of two rigid bodies, the base and theplatform, connected by a number of other rigid bodies, called legs, that areanchored via spherical joints. It is possible to prove that the maximal numberof legs of a mobile pod, when finite, is 20. In 1904, Borel designed a technique toconstruct examples of such 20-pods, but could not constrain the legs to have baseand platform points with real coordinates. We show that Borel’s constructionyields all mobile 20-pods, and that it is possible to construct examples whereall coordinates are real.

Keywords: icosapods, line-symmetric motion, body-bar framework,spectrahedra2000 MSC: 14L35, 70B15, 14P10, 53A17

Introduction

A multipod is a mechanical linkage consisting of two rigid bodies, called thebase and the platform, and a number of rigid bodies, called legs, connectingthem. Each leg is attached to base and platform with spherical joints (seeFigure 1), so platform points are constrained to lie on spheres — the centerof each sphere is then the base point connected to the respective leg. If theplatform can move respecting the constraints imposed by the legs we say thatthe multipod is mobile.

Note that multipods are also studied within Rigidity Theory as so-calledbody-bar frameworks [1], as two rigid bodies (platform and base) are connectedby multiple bars (legs). Mobile multipods correspond to flexible body-bar frame-works, whose study is of great practical interest e.g. for protein folding [2].

We can model the possible configurations of a multipod using direct isome-tries of R3, by associating to every configuration the isometry, which maps areference configuration into the respective one. At this point one can consider amotion, namely a one-dimensional set of direct isometries, and try to constructa multipod moving according to this motion. This approach has a long history;there are motions allowing multipods with infinitely many legs, but among thosethat allow only a finite number of legs, the maximal number is 20. This wasproved by Schoenflies, see Remark 2.6 below. Borel proposed a construction for

Preprint submitted to Elsevier December 12, 2016

icosapods (namely multipods with 20 legs) leading to linkages whose motion isline-symmetric, i.e. whose elements are involutions, namely rotations by 180◦

around a line (see Figure 1).

`

Figure 1: Base points (pink) and platform points (yellow) of an icosapod. Notice that forevery leg there is a symmetric one obtained by rotating the first one by 180◦ around theaxis `. The symmetry reverses the role of base and platform points.

This paper provides two results on icosapods. First, we show that all mobileicosapods (with very mild restrictions) are instances of Borel’s construction.Second, we exhibit a mobile icosapod that is an instance of Borel’s construction(Borel, in fact, obtained equations for the base and platform points of a mobileicosapod, but could not prove the existence of solutions in R3). The second partis closely related to the theory of quartic spectrahedra, which has been studiedin [3].

Section 1 provides an historical overview of line-symmetric motions. In Sec-tion 2 we set up the formalism and the objects that are needed for our approachto the problem; in particular we recall a compactification of the group of directisometries that has already been used by the authors to deal with problems onmultipods, and we show how the constraints imposed by legs can be interpretedas a duality between the space of legs and the space of direct isometries. InSection 3 we use these tools to prove that, under certain generality conditions,mobile icosapods admit line-symmetric motions. In Section 4 we show how itis possible to construct example of mobile icosapods employing results in thetheory of quartic spectrahedra.

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1. Review on line-symmetric motions

Krames [4] studied special one-parametric motions, obtained by reflectingthe moving system ς in the generators of a ruled surface in the fixed system Σ.This ruled surface is called the base surface for the motion. He showed someremarkable properties of these motions (see [4]), which led him to name themSymmetrische Schrotung (in German). This name was translated to Englishas symmetric motion by Tölke [5], Krames motion or line-symmetric motionby Bottema and Roth [6, page 319]. As each Symmetrische Schrotung hasthe additional property that it is equal to its inverse motion (cf. Krames [7,page 415]), it could also be called involutory motion. In this paper we use thename line-symmetry as it is probably the most commonly used term in today’skinematic community.

Further characterizations of line-symmetric motions (beside the cited one ofKrames [4]) where given by Tölke [5], Bottema and Roth [6, Chapter 9, § 7],Selig and Husty [8] and Hamann [9]. If one uses the so-called Study parameters(e0 : e1 : e2 : e3 : f0 : f1 : f2 : f3) to describe isometries, then it is possible tocharacterize line-symmetric motions algebraically in the following way. Givensuch a motion, there always exist a Cartesian system of coordinates, or frame(o;x, y, z) for the moving system ς and a Cartesian frame (O;X,Y, Z) for thefixed system Σ so that e0 = f0 = 0 holds for the elements of the motion. Withthis choice of coordinates, the latter are rotations by 180◦ around lines; theStudy coordinates (e1 : e2 : e3 : f1 : f2 : f3) of these isometries coincide withthe Plücker coordinates of the lines.

1.1. Historical results on line-symmetric motions with spherical pathsIn this paper we study line-symmetric motions that are solutions to the still

open problem posed by the French Academy of Science for the Prix Vaillantof the year 1904 (cf. [10]): "Determine and study all displacements of a rigidbody in which distinct points of the body move on spherical paths." Borel andBricard were awarded the prize for their papers [11] and [12] containing partialsolutions, and therefore this is also known as the Borel Bricard (BB) problem.

1.1.1. Krames’s resultsKrames [7, 13, 14] studied some special motions already known to Borel and

Bricard in more detail and stated the following theorem [7, Satz 6]:

Theorem 1.1. For each line-symmetric motion, that contains discrete, 1 or2-dimensional spherical paths, the set f of points with spherical trajectories iscongruent (direct isometry) to the set F of corresponding sphere centers.

Moreover Krames noted in [7, page 409] that1 ". . .most of the solutions givenby Borel and Bricard are line-symmetric motions. In each of these motions bothgeometers detected this circumstance by other means, without using the abovementioned result" (Theorem 1.1). In the following we will take a closer look atthe papers [11, 12], which shows that the latter statement is not entirely correct.

1The following extract as well as Theorem 1.1 has been translated from the original Germanby the authors.

3

1.1.2. Bricard’s resultsBricard studied these motions in [12, Chapitre VIII]. His first result in this

context (see end of [12, § 32, page 70]) reads as follows (adapted to our notation):Dans toutes les solutions auxquelles on sera conduit, les figures liées F et f

seront évidemment égales et semblablement placées par rapport aux deux trièdres(O;X,Y, Z) et (o;x, y, z).

In the remainder of [12, Chapitre VIII] he discussed some special cases,which also yield remarkable results, but he did not give further information onthe general case.

1.1.3. Borel’s resultsBorel discussed in [11, Case Fb] exactly the case e0 = f0 = 0 and he proved

in [11, Case Fb1] that in general a set of 20 points are located on sphericalpaths but without giving any result on the reality of the 20 points. Moreoverhe studied two special cases in Fb2 and Fb3.

Borel did not mention the geometric meaning of the assumption e0 = f0 = 0.He only stated at the beginning of case F [11, page 95] that the moving frame(o;x, y, z) is parallel to the frame obtained by a reflection of the fixed frame(O;X,Y, Z) in a line. This corresponds to the weaker assumption e0 = 0. Headded that this implies the same consequences as already mentioned in [11,page 47, case C], which reads as follows (adapted to our notation):

. . . dans le cas où les trièdres sont symétriques par rapport à une droite, sideux courbes sont représentées par des équations identiques, l’une en X,Y, Z,l’autre en x, y, z, elles sont symétriques par rapport à cette droite.

But Borel did not mention, neither in case Fb1 nor in his conclusion section,that f with #f = 20 is congruent to F (contrary to other special cases e.g. Fb3,where the congruence property is mentioned explicitly.)

1.2. Review of line-symmetric self-motions of hexapodsWe denote the platform points of the i-th leg in the moving system ς by pi

and its corresponding base points in the fixed system Σ by Pi.For a generic choice of the geometry of the platform and the base as well

as the leg lengths di the hexapod can have up to 40 configurations. Undercertain conditions it can also happen that the direct kinematic problem hasno discrete solution set but an n-dimensional one with n > 0. Clearly theseso-called self-motions of hexapods are solutions to the BB problem.

In practice hexapods appear in the form of Stewart-Gough manipulators,which are 6 degrees of freedom parallel robots. In these machines the leg lengthscan be actively changed by prismatic joints and all spherical joints are passive.

Moreover a hexapod (resp. Stewart-Gough manipulator) is called planar ifthe points p1, . . . , p6 are coplanar and also the points P1, . . . , P6 are coplanar;otherwise it is called non-planar. In the following we review those papers whereline-symmetric self-motions of hexapods are reported.

1.2.1. Non-planar hexapods with line-symmetric self-motionsLine-symmetric motions with spherical paths already known to Borel [11]

and Bricard [12] (and also discussed by Krames in [7, 14]) were used by Hustyand Zsombor-Murray [15] and Hartmann [16] to construct examples of (planarand non-planar) hexapods with line-symmetric self-motions.

4

Point-symmetric hexapods with congruent platform and base possessing line-symmetric self-motions were given in [17, Theorem 11]. Further non-planarhexapods with line-symmetric self-motions can be constructed from overcon-strained pentapods with a linear platform [18].

1.2.2. Planar hexapods with line-symmetric self-motionsAll self-motions of the original Stewart-Gough manipulator were classified

by Karger and Husty [19]. Amongst others they reported a self-motion with theproperty e0 = 0 (see [19, page 208, last paragraph]), "which has the propertythat all points of a cubic curve lying in the plane . . . and six additional pointsout of this plane have spherical trajectories. This seems to be a new case of aBB motion, not known so far." Based on this result, Karger [20, 21] presenteda procedure for computing further "new self-motions of parallel manipulators"of the type e0 = 0, where the points of a planar cubic c have spherical paths.

Another approach was taken by Nawratil in his series of papers [22, 23,24, 25], by determining the necessary and sufficient geometric conditions forthe existence of a 2-dimensional motion such that three points in the xy-planeof ς move on three planes orthogonal to the XY -plane of Σ (3-fold Darbouxcondition) and two planes orthogonal to the xy-plane of ς slide through two fixedpoints located in the XY -plane of Σ (2-fold Mannheim condition). It turned outthat all these so-called type II Darboux-Mannheim motions are line-symmetric.Moreover a geometric construction of a 12-parametric set of planar Stewart-Gough platforms (cf. [25, Corollary 5.4]) with line-symmetric self-motions wasgiven. It was also shown that the algorithm proposed by Karger in [20, 21]yields these solutions.

While studying the classic papers of Borel and Bricard for this historicalreview we noticed that the solution set of the BB problem mentioned in thelast two paragraphs was already known to these two French geometers; cf. [11,Case Fb3] and [12, Chapter V] (already reported by Bricard in [26, page 21]).But in contrast to the above listed approaches (of Karger and Nawratil) bothof them assumed that the motion with spherical trajectories is line-symmetric.Each of them additionally discovered one more property:

• Borel pointed out that there exist further 8 points (all 8 can be real) withspherical trajectories. This set of points splits in four pairs, which are sym-metric with respect to the carrier plane of the cubic c.

• Bricard showed the following: If we identify the congruent planar cubics ofthe platform and the base, i.e. c = C, then the tangents in a correspondingpoint pair P and p with respect to c = C intersect each other in a point ofthe cubic c = C (P and p form a so-called Steinerian couple).

Bricard communicated his result (published in [26, page 21]) to Duporcq,who gave an alternative reasoning in [27], which sank into oblivion over thepast 100 years. Only a footnote in the conclusion section of Borel’s work [11]points to Duporcq’s proof (but not to the original work of Bricard [26]), whichis based on the following remarkable motion (see Figure 2):

Let P1, . . . , P6 and p1, . . . , p6 be the vertices of two complete quadrilaterals,which are congruent. Moreover the vertices are labelled in a way that pi isthe opposite vertex of Pi for i ∈ {1, . . . , 6}. Then there exist a two-parametricline-symmetric motion where each pi moves on a sphere centered at Pi.

5

P1 P2 P3

P5 P4

P6

p4 p5 p6

p2 p1

p3

Figure 2: Illustration of Duporcq’s complete quadrilaterals.

It can be easily checked that this configuration of base and platform pointscorresponds to an architecturally singular hexapod (e.g. [28] or [29]). As archi-tecturally singular manipulators are redundant we can remove any leg — w.l.o.g.we suppose that this is the sixth leg — without changing the direct kinematicsof the mechanism. Therefore the resulting pentapod P1, . . . , P5 and p1, . . . , p5

also has a two-parameter, line-symmetric, self-motion.Note that this pentapod yields a counter-example to Theorem 4.2 of [30] and

as a consequence also the work [31] is incomplete as it is based on this theorem.For the erratum to [30] please see [32] and the addendum to [31] is given in [33].

Remark 1.2. If we assume for this pentapod that the line [P1P2] is the idealline of the fixed plane (so [p4p5] is the ideal line of the moving plane) then weget exactly the conditions found in [22], which the points and ideal points ofthe plane’s normals must fulfill in order to get a type II Darboux-Mannheimmotion. Note that P2, P3 can also be complex conjugates (so p5, p6 are complexconjugate too).

1.2.3. Computer search for mobile hexapodsIn [34], Geiss and Schreyer describe a rather non-standard way to find mobile

hexapods. They set up an algebraic system of equations equivalent to mobility,and then try random candidates, with coordinates in a finite field of smallsize, by computer. After collecting statistical data indicating the existence ofa family of real mobile hexapods, they try a (computationally more expensive)lifting process in the most promising cases. The method is extremely powerfuland could still be used for finding new families of mobile hexapods. However,the family reported in [34] can be seen as an instance of Borel’s family Fb1.The line symmetry is not apparent because the method starts by guessing 6legs, which may not form a line symmetric configuration; only if one adds theremaining 14 legs, or at least all real legs among them, one obtains a symmetricconfiguration. The question posed in Problem 5 in [34] is easily answered fromthis viewpoint: two legs have the same lengths because they are conjugated byline symmetry.

2. Isometries, legs and bond theory

This section illustrates the concepts and techniques that will be needed tocarry on our analysis. From now on, by an n-pod we mean a triple (~p, ~P , ~d) where~p, ~P are n-tuples of vectors in R3 (respectively, platform and base points), and~d is an n-tuple of positive real numbers. Multipods with 20 legs are calledicosapods.

6

Remark 2.1. The description of admissible configurations that we adopt allowsindependent choices for two coordinate systems, one for the base and one for theplatform: this is why also the leg lengths have to be included in the definition ofa pod. Moreover, we do not require that the identity belongs to the admissibleconfigurations.

2.1. Isometries and leg spaceThroughout this paper we use a compactification of the group SE3 of di-

rect isometries that was introduced in [35] and in [36]. We briefly recall thisconstruction.

Consider a point a lying on a sphere centered at b and with radius d, theequation expressing this coincidence can be written,

‖a− b‖2 − d2 = 0.

Expanding this gives (〈a, a〉+ 〈b, b〉 − d2

)− 2 〈a, b〉 = 0,

where 〈·, ·〉 is the Euclidean scalar product. This can be written as a scalarproduct of two 5-vectors, one containing information on the sphere the otheronly information about the point,

(−2bt, (〈b, b〉 − d2), 1

) a1〈a, a〉

= 0.

Now a direct isometry σ, of R3 can be described by a pair (M,y), whereM is an orthogonal matrix with det(M) = 1 and y is the image of the originunder the isometry. If the point a remains on the sphere after the action of σ,then we have, ‖σ(a)− b‖2 − d2 = 0. In terms of the 5-vectors we can write thiscondition as,

(−2b, (〈b, b〉 − d2), 1

)M y 00 1 0

2xt r 1

a1〈a, a〉

= 0,

where the matrix represents the action of the isometry and we have definedx := −M ty = −M−1y and r := 〈x, x〉 = 〈y, y〉. Notice that this determines a5-dimensional representation of the group SE3, displaying the direct isometriesas a subgroup of the group of conformal transformations of space.

If a, b ∈ R3 and d ≥ 0, then the condition ‖σ(a)− b‖2 − d2 = 0 on a directisometry σ is linear, and in particular has the following form:(

〈a, a〉+ 〈b, b〉 − d2)h+ r − 2 〈a, x〉 − 2 〈y, b〉 − 2 〈Ma, b〉 = 0, (1)

where the homogenizing variable h has been included. We will usually refer toEquation (1) as the sphere condition imposed by (a, b, d).

So, if we take coordinates m11, . . . ,m33, x1, x2, x3, y1, y2, y3 and r, h in P16C,

a direct isometry defines a point in projective space satisfying h 6= 0 and

MM t = M tM = h2 · idR3 , det(M) = h3,

M ty + hx = 0, Mx+ hy = 0,

〈x, x〉 = 〈y, y〉 = r h.

(2)

7

The equations in (2) define a variety X in P16C, whose real points satisfying h 6= 0

are in one to one correspondence with the elements of SE3. A direct calculationshows that X is a variety of dimension 6 and degree 40.

Given an n-pod Π = (~p, ~P , ~d), we can consider its configuration space, namelythe set of direct isometries σ in SE3 such that ‖σ(pi)− Pi‖2 − d2

i = 0 for alli ∈ {1, . . . , n}. The fact that Equation (1) is linear in the coordinates of P16

Cmeans that the configuration space of Π can be compactified as the intersectionof X with a linear space Λ, determined by the n conditions imposed by itslegs; we denote such variety by KΠ. We say that the pod Π is mobile if theintersection KΠ(R) ∩ {h 6= 0} — where we denoted by KΠ(R) the real pointsof KΠ — has (real) dimension greater than or equal to one. Notice that if Π ismobile, then KΠ has (complex) dimension greater than or equal to one.

Notation. From now on, by the expression mobility one icosapod we mean a20-pod with mobility one which cannot be obtained by removing legs from amultipod with the same configuration set.

Remark 2.2. For reasons of completeness it should be noted that system inEquation (2) was already used by Mourrain [37, page 293] to prove that a non-mobile pod has at most 40 configurations. Similar systems of equations werealso used by Lazard [38, page 179] and Wampler [39, Equation (2)] for the sametask.

For our purposes, it is useful to introduce another 16-dimensional projectivespace, playing the role of a dual space of the one containing X, where the dualityis given by a bilinear version of Equation (1). We start by introducing a newquantity, called corrected leg length, defined as l := 〈a, a〉 + 〈b, b〉 − d2, so thatEquation (1) becomes

l h+ r − 2 〈a, x〉 − 2 〈y, b〉 − 2 〈Ma, b〉 = 0.

We think of the points a = (a1, a2, a3) and b = (b1, b2, b3) as points in P3Cby

introducing two extra homogenization coordinates a0 and b0. In this way, thepair (a, b) can be considered as a point in the Segre variety Σ3,3

∼= P3C× P3

C; the

latter is embedded in P15C, where we take coordinates {zij} so that the points

of Σ3,3 satisfy zij = ai bj for some (a0 : a1 : a2 : a3), (b0 : b1 : b2 : b3) ∈ P3C. If

we homogenize Equation (1) with respect to the coordinates {zij} and l, thenwe get

l h+ z00r − 2 (z10x1 + z20x2 + z30x3)−

− 2 (z01y1 + z02y2 + z03y3)− 2

3∑i,j=1

mij zij = 0(3)

Notice that the left hand side of Equation (3) is a bilinear expression in the co-ordinates (h,M, x, y, r) and in the coordinates (z, l). We denote this expressionby BSC (for bilinear sphere condition). Hence, if we denote by P̌16

Cthe projec-

tive space with coordinates (z, l), then we obtain a duality between P16C

and P̌16C

sending a point (h0,M0, x0, y0, r0) ∈ P16C

to the hyperplane in P̌16C

of equationBSC(h0,M0, x0, y0, r0, z, l) = 0, and a point (z0, l0) ∈ P̌16

Cto the hyperplane

in P16C

of equation BSC(h,M, x, y, r, z0, l0) = 0.

8

Remark 2.3. Suppose that (z0, l0) ∈ P̌16C

belongs to the cone Y with vertex(0 : · · · : 0 : 1) over the Segre variety Σ3,3 — namely (z0)ij = ai bj for some(a0 : a1 : a2 : a3) and (b0 : b1 : b2 : b3). Suppose furthermore that a0 = b0 = 1and a2

1 + a22 + a2

3 + b21 + b22 + b23− l0 ≥ 0. Then from the definition of BSC we seethat the hyperplane BSC(h,M, x, y, r, z0, l0) = 0 in P16

Cis the same hyperplane

defined by Equation (1) where we take a = (a1, a2, a3), b = (b1, b2, b3) andd =

√a2

1 + a22 + a2

3 + b21 + b22 + b23 − l.Remark 2.3 indicates that the cone Y in P̌16

Cplays a sort of dual role to the

one of the compactification X in P16C, and we will exploit this in our arguments.

Since the Segre variety Σ3,3 has dimension 6 and degree 20, we see that Y hasdimension 7 and degree 20.

Definition 2.4. Let C ⊆ X be a curve. We define the leg set LC as the setof all points (z, l) ∈ Y such that the BSC — instantiated at (z, l) — holdsfor all points in C. The leg set is a compactification of the set of all triples(a, b, d) ∈ R3 × R3 × R≥0 such that the image of a under any point in C lyingin the image of SE3 (hence considered as an isometry) has distance d from b.

Proposition 2.5. Let C ⊆ X be a curve. If LC has only finitely many complexpoints, then its cardinality is at most 20. If LC has exactly 20 points, then thelinear span of LC in P̌16

Cis a projective subspace of dimension 9.

Proof. By construction LC is defined by linear equations as a subset of Y ; inother words LC = Y ∩ span(LC). Then the statement follows from generalproperties of linear sections of projective varieties. In fact, any linear subspaceof codimension less than 7 intersects Y in a subvariety of positive dimension,hence dim

(span(LC)

)≤ 9. A general linear subspace of dimension 9 intersects Y

in deg(Y ) = 20 points. In order to prove that dim(span(LC)

)= 9 when LC

has 20 points we assume the contrary, that dim(span(LC)

)< 9. Then we

take a general linear superspace Λ of span(LC) of dimension 9. It intersects Yagain in 20 points, which must coincide with LC . On the other hand, if theHilbert series of Y is P (t)

(1−t)16 , then the Hilbert series of Λ∩ Y is P (t)(1−t)9 , but this

contradicts the fact that Λ∩Y = LC is contained in a linear space of dimensionstrictly smaller than 9.

In order to show the maximal number of 20 intersections can be achieved inProposition 2.5, we need to find a curve C ⊆ X such that the bilinear sphereconditions of its points define a linear subspace in P̌16

Cof dimension 9. This is

equivalent to asking dim(span(C)

)= 15−9 = 6. We will deal with this problem

in Section 3.

Remark 2.6. The proof of Proposition 2.5 also gives a simple alternative prooffor the number of solutions (over C) of the spatial Burmester problem, whichreads as follows: Given seven poses ς1, . . . , ς7 of a moving system ς, determineall points of ς, that are located on a sphere in all seven poses.

The given poses correspond to seven points in X, which span in the generalcase a P6

C. Therefore its dual space is of dimension 9, and so intersects Y in

exactly 20 points.This number was first computed by Schoenflies in [40, page 148] and con-

firmed by Primrose (see [41, footnote 3]) as well as by Wampler et al. [42,Section 5]. We conclude noticing that the first solution of the spatial Burmester

9

problem using an approach based on polynomial systems was presented by In-nocenti [43], who also gave an example with 20 real solutions.

2.2. Bond theoryThe second ingredient for our arguments is bond theory, a technique for

analyzing mobile multipods that was introduced and developed (in the formwe need here) in [36]. If KΠ is the configuration curve of a multipod Π ofmobility one, then the bonds of Π are defined as the intersections of KΠ withthe hyperplane H =

{h = 0

}. Note that such intersections always arise in

conjugate complex pairs since KΠ is a real algebraic variety and has no realpoints.

Recall, that the elements of SE3 are in 1-to-1 correspondence with the realpoints of the variety X so long as h 6= 0. So the bonds are points in the closureof SE3 in this model of the group. The set B = H ∩X has dimension 5, and itspoints, although they do not represent isometries, still have geometric meaningas conditions imposed on the legs: we mean that if the configuration set KΠ

passes through a point in B than base and platform points of Π must satisfy acertain condition. The variety B can be partition into five subsets, which differby the condition imposed on base and platform points of multipods. Detail ofthese subsets can be found in [36], here we simply list the possibilities. We justpoint out that we are not going to use the properties of inversion and similaritypoints, we report them here only for the sake of completeness.

vertex the only real point in B; it is never contained in a configuration curve.

inversion points if a multipod Π has a configuration set passing through aninversion point, then there exists two directions L and R in R3 such thatif we project the base of Π orthogonally along L and the platform of Πorthogonally along R, we obtain two tuples in the plane that correspondvia an inversion (depending on the boundary point).

similarity points if a multipod Π has a configuration set passing through asimilarity point, then there exists two directions L and R in R3 such thatif we project the base of Π orthogonally along L and the platform of Πorthogonally along R, we obtain two tuples in the plane that correspondvia a similarity (depending on the boundary point).

butterfly points these correspond to a pair of lines in R3, one for the baseand one for the platform; a multipod Π whose configuration set passesthrough a butterfly point either has the base point on the base line or theplatform point on the platform line.

collinearity points these correspond to lines; if a multipod Π has a configu-ration set passing through a collinearity points, then either its platformpoints or its base points are collinear.

A mobile icosapod cannot have butterfly bonds (namely bonds that arebutterfly points), nor can it have collinearity bonds as this would imply thatthere are at least 10 legs with collinear base points or platform points. Thenall points of the line carrying those base or platform points would be base orplatform points for a multipod with the same configuration set. We will notconsider these cases in our discussion.

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Inversion points are smooth points of X, and their tangent spaces are con-tained in H. Consequently every motion passing an inversion point is tangentto the hyperplane H at this point. In particular, if a multipod Π of mobilityone has only inversion bonds, the degree of its configuration curve C is twicethe number of inversion bonds: the degree can be computed by intersecting Cwith H, and there we see only pairs of double intersections.

Consider the projection P16C

99K P9Ckeeping only the coordinates h and mij

for i, j ∈ {1, 2, 3}. The real points of the open subset defined by h 6= 0 in theimage of X is the group variety SO3, and the image itself is a subvariety Xm ofdegree 8 that is isomorphic to the Veronese embedding of P3

Cby quadrics. This

follows from the bijection between points (e0 : e1 : e2 : e3) ∈ P3Rand orthogonal

matrices (see [44, Section 4.5]) given by

(e0 : e1 : e2 : e3)l

1e20+e21+e22+e23

e20+e21−e22−e

23 2e1e2−2e0e3 2e0e2+2e1e3

2e1e2+2e0e3 e20−e21+e22−e

23 −2e0e1+2e2e3

−2e0e2+2e1e3 2e0e1+2e2e3 e20−e21−e

22+e23

. (4)

In fact, if we consider the Veronese variety that is the image of the morphism

P3C

−→ P9C

(e0 : e1 : e2 : e3) 7→ (e20 : e2

1 : e22 : e2

3 : e0 e1 : e0 e2 : e0 e3 : e1 e2 : e1 e3 : e2 e3)

and we apply the projective automorphism given by the matrix

1 1 1 1 0 0 0 0 0 01 1 −1 −1 0 0 0 0 0 00 0 0 0 0 0 −2 2 0 00 0 0 0 0 2 0 0 2 00 0 0 0 0 0 2 2 0 01 −1 1 −1 0 0 0 0 0 00 0 0 0 −2 0 0 0 0 20 0 0 0 0 −2 0 0 2 00 0 0 0 2 0 0 0 0 21 −1 −1 1 0 0 0 0 0 0

, (5)

then a direct computation shows that we obtain Xm. The coordinates of P3C

are called Euler parameters and denoted by e0, e1, e2, e3. The center of theprojection P16

C99K P9

Cintersects X in the union of the sets of similarity points,

collinearity points and the vertex.

2.3. The subvariety of involutionsWe focus our attention on a particular subvariety of X, the compactification

of the set of involutions in SE3. Involutions in SE3 are rotations of 180◦ arounda fixed axis, so their compactification — which we will denote by Xinv — is a 4-dimensional subvariety of X, because the family of lines in R3 is 4-dimensional.One reason why involutions are particularly useful in the creation of mobile podsis that if p and P are a platform and a base point of a pod Π, and σ ∈ KΠ is aninvolution in the configuration space of Π, then this means that ‖σ(p)− P‖ = d,where d is the distance between p and P ; on the other hand, since σ is aninvolution we have ‖σ(P )− p‖ = d. This means that if all isometries in theconfiguration space of Π are involutions, then we can swap the roles of baseand platform points and obtain “for free” new legs not imposing any furtherrestriction to the possible configurations of Π.

11

If σ = (M,y) is an involution, then M = M t, that is to say M is symmetric,and y = x. Hence we can consider the subvariety{

(h : M : x : y : r) ∈ X : M = M t and x = y}.

One verifies that this subvariety has two irreducible components, namely theisolated point corresponding to the identity and another one of dimension 4,which is cut out by a further linear equation, namely m11 +m22 +m33 +h = 0.

Definition 2.7. The subvariety of X defined by the equations M = M t andx = y and m11 +m22 +m33 + h = 0 is denoted Xinv.

3. Mobile icosapods are line-symmetric

In this section we will show that if an icosapod of mobility one admits an ir-reducible configuration curve, then its motion is line-symmetric (Theorem 3.10).We start by translating this concept into our formalism. Recall from [36, Sec-tion 2.2] that the group SE3 acts on its compactification X: every isometryin SE3 determines a projective automorphism of P16

Cleaving X invariant.

Definition 3.1. Let C ⊆ X be a curve. Then C is called an involutory motionif C ⊆ Xinv. The curve C is called a line-symmetric motion if there exists anisometry τ such that the automorphism associated to τ maps C inside Xinv.

From Proposition 2.5 we know that the configuration curve of an icosapodspans a linear subspace of dimension 6. To get an overview of possible examplesof irreducible curves C ⊆ X with dim

(span(C)

)= 6, we consider the projection

P16C

99K P9Cdescribed at the end of Subsection 2.2. Let Cm ⊆ Xm be the

projection of C, which can be either a point or a curve. It is possible to prove(see [45]) that if Cm is a point, then there exists a multipod with infinitelymany legs admitting C as configuration set. Since we are interested in podswith finitely many legs, from now we suppose that Cm is a curve. Let Ce ⊆ P3

Cbe its isomorphic preimage under the Veronese map.

Proposition 3.2. If C ⊆ X is an irreducible curve such that dim(span(C)

)=

6, then Ce is either planar or a twisted cubic.

Proof. The projection of span(C) in P9Cis a linear subspace of dimension at

most 6. Hence the ideal of Cm contains at least 3 linear independent linearforms. Hence the ideal of Ce contains at least 3 linear independent quadraticforms. If any of such quadratic form is reducible, namely splits into the unionof two planes, the statement follows; thus from now on we can suppose that allof them are irreducible. Then the intersection of the zero sets of two of thesequadrics is a quartic curve D such that Ce ⊆ D. It cannot happen that Ce = D,because this would contradict the fact that there are three independent quadricspassing through Ce. Therefore the degree of Ce can only be 1, 2 or 3. If thedegree is 1 or 2, then Ce is planar; if the degree is 3, then Ce is either planaror a twisted cubic, because by construction Ce is irreducible. The propositionthen follows.

From now on, since we aim for a result on mobile icosapods, we will considercurves C allowing exactly 20 legs, that is satisfying the following condition:{

C is an irreducible real curve with real points,LC consists of exactly 20 real finite points. (†)

12

Here by real “finite” points we mean that their z00-coordinates are not zero; inother terms, such points determine pairs of base and platform points in R3 (andnot at infinity).

Remark 3.3. Notice that condition (†) does not comprise every mechanical de-vice that one could name a “mobile icosapod”: there may exist a device admit-ting infinitely many complex points in its leg space, of which only 20 are realand finite; we believe that such situation cannot occur, but we were not ableto provide an argument for this. Still, we believe that condition (†) is a goodcompromise because it will allow a uniform treatment of the topic.

Remark 3.4. Notice that condition (†) implies that dim(span(C)

)= 6. More-

over, from Section 2.2 it follows that C does not pass through any butterfly orcollinearity point.

Lemma 3.5. Suppose that C ⊆ X satisfies condition (†). Then Ce cannot bea cubic (neither a twisted cubic, nor a plane cubic).

Proof. Suppose that C satisfies condition (†) and Ce is a twisted cubic. Then thecurve Cm, isomorphic to Ce under the Veronese embedding, is a rational normalsextic, and therefore spans a linear space of dimension 6. This means that theprojection C −→ Cm is a projective isomorphism — since by Remark 3.4 alsothe curve C spans a linear space of dimension 6. This forces the center of theprojection to be disjoint from span(C). By Section 2.2, this implies that C doesnot admit similarity or collinearity points. Recall from Remark 3.4 that thecurve C does not admit butterfly points, so the only ones left are the inversionpoints. However, since the degree of C is twice the number of inversion points,we would get 3 of them, and this is not possible, since they occur in conjugatepairs.

Now, suppose that Ce is a planar cubic. Then the curve Cm spans a linearspace of dimension 5. This means that the center of the projection C −→ Cm

intersects span(C) in a single point. Since similarity points occur in pairs, asremarked at the beginning of Section 2.2, such intersection cannot be a similaritypoint; as before, Remark 3.4 ensures that the curve C does not admit butterflyor collinearity points, so C has only inversion points. Hence we can conclude asin the previous case.

We focus therefore on curves C satisfying (†) such that Ce is planar of degreedifferent from 3. First we rule out the case when Ce is a line or a conic.

Lemma 3.6. Suppose that C ⊆ X satisfies condition (†). Then Ce cannot bea line.

Proof. If Ce is a line the corresponding motion can only be a Schoenflies motion2.These motions with points moving on spheres where previously studied by Hustyand Karger in [46]. It is not difficult to see that no discrete solution can exist,as any leg can be translated along the axis of the Schoenflies motion withoutrestricting the self-motion. Therefore we always end up with an ∞-pod.

2These motions can be composed by a rotation about a fixed axis and an arbitrary trans-lation.

13

Lemma 3.7. Suppose that C ⊆ X satisfies condition (†). Then Ce cannot bea conic.

Proof. Assume that Ce is a conic. The bilinear spherical condition BSC deter-mines a subspace P̌9

C⊆ P̌16

Cas dual to the linear projection P16

C99K P9

C. By

a direct inspection of Equation (3) one notices that the subvariety Y∞ ⊆ Ycomposed of those pairs (a, b) of points with a0 = b0 = 0 (namely legs forwhich both the base and the platform point are at infinity) is contained in P̌9

C.

The dimension of Y∞ is 5 and its degree is 6, since it is a cone over the Segrevariety P2

C× P2

C.

Consider now the set of linear forms in P9Cvanishing on Cm: this is a vector

space of dimension 5, since it is isomorphic to the vector space of all quadraticforms on P3

Cthat vanish along Ce. In this way we get a linear space P̌4

C⊆ P̌9

C.

The intersection P̌4C∩ Y∞ ⊆ LC is non-empty and is in general constituted of

6 points. Thus the number of real legs not at infinity is at most 14, and thiscontradicts the assumption that LC has 20 real finite points.

For the remaining cases, when Ce is a planar curve of degree greater than orequal to 4, we want to prove that there exists an isometry τ such that the imageof C under the corresponding projective automorphism is contained in Xinv.Recall from Section 2.2 that we denoted e0, . . . , e3 the coordinates of the P3

Cwhere Ce lives. We may assume without loss of generality that e0 = 0 holds forthe points of Ce; this can be achieved by a suitable rotation of the coordinateframe of the platform — specifically by acting on C with a suitable rotation —since by assumption Ce is planar. In terms of the coordinates of X, this meansthat we can apply an automorphism of P16

Cinduced by an isometry so that the

points of C satisfy mij = mji and m11 +m22 +m33 + h = 0: this follows fromthe relations between the variables (h : M) and the variables (e0 : e1 : e2 : e3)(see in particular Equations (4) and (5) when e0 = 0).

We can use the assumption e0 = 0 in order to simplify the embedding. LetX1 be the intersection of X with the linear space{

(h : M : x : y : r) ∈ P16C

: mij = mji and m11 +m22 +m33 + h = 0}.

Notice that X1 is the set-theoretical preimage of the locus {e0 = 0} under theprojectionX 99K P3

C, where such projection is obtained composing the projection

to P9Con the (h : M)-coordinates with the inverse of the Veronese embedding

P3C−→ P9

C. We project X1 from the point (0 : · · · 0 : 1) in P16

C— the only

singular point of X of order 20 — obtaining a subvariety X2 of P15C, which is

actually contained in a linear subspace isomorphic to P11C

because of the linearequations imposed on X1.

If we express the coordinates h and mij for i, j ∈ {1, 2, 3} in terms of theEuler coordinates e1, e2, e3, and apply the coordinate change

pi = xi + yi, qi = xi − yi, for i ∈ {1, 2, 3},

then we obtain a map from X2 to the weighted projective space PC(~1,~2) (see[47, Example 10.27] for a reference for weighted projective space). Here ~1 =(1, 1, 1) and ~2 = (2, 2, 2, 2, 2, 2) and we take coordinates e1, e2, e3 of weight 1and p1, p2, p3, q1, q2, q3 of weight 2. The image of X2 is a weighted projective

14

variety Z ⊆ PC(~1,~2) of dimension 5 defined by the equations

e1 p1 + e2 p2 + e3 p3 = p1 q1 + p2 q2 + p3 q3 =

e1 q2 − e2 q1 = e1 q3 − e3 q1 = e2 q3 − e3 q2 = 0,

as a direct computation using computer algebra confirms. Therefore, for acurve C ⊆ P16

Cfor which Ce is planar and satisfies e0 = 0 we get a map C −→

Cz ⊆ Z that is the composition of a projection, a linear change of variables anda Veronese map.Remark 3.8. At the beginning of the section we pointed out that isometriesdetermine automorphisms of P16

Cleaving X invariant. Notice that the auto-

morphisms corresponding to translations leave X1 invariant, since its equationscomprise only the rotational part of isometries. Therefore translations also acton Z.

Lemma 3.9. Suppose that C ⊆ X satisfies condition (†). Let Cz ⊆ Z be theimage of the curve C under the previously defined maps. Then there exists atranslation of the platform such that the corresponding automorphism maps Cz

to a curve C ′z whose points satisfy q1 = q2 = q3 = 0.

Proof. From the discussion so far we may infer that deg(Ce) > 3. We are espe-cially interested in the q-vector, so let W ⊆ PC(1, 1, 1, 2, 2, 2) be the projectionof Z to the e and q-coordinates and let Cw be the image of Cz under suchprojection. The set W has dimension 4, and its equations are

e1 q2 − e2 q1 = e1 q3 − e3 q1 = e2 q3 − e3 q2 = 0. (6)

By a direct inspection of the map C −→ Cw one notices that forms of weighteddegree 2 on Cw correspond to linear form on C. It follows that the vector spaceof weighted degree 2 forms on Cw has dimension at most 7. There are 9 formsof weighted degree 2 on PC(1, 1, 1, 2, 2, 2) and they are all linear independent asforms on W because the latter is defined by equations of weighted degree 3.Hence Cw satisfies at least 2 equations E1 = E2 = 0 of weighted degree 2.

By construction, the polynomials Ei are of the form Ei = Li (~q) + Qi (~e),where Li is linear and Qi is quadratic. Notice that L1 (~q)L2 (~e)− L1 (~e)L2 (~q)vanishes on W , because it is a multiple of the polynomials in Equation (6).Therefore on Cw we have

E1 (~e, ~q) L2 (~e)− E2 (~e, ~q) L1 (~e) = Q1 (~e) L2 (~e)−Q2 (~e) L1 (~e) = 0.

The latter is a cubic equation only in the variables ~e, thus it is satisfied by Ce.On the other hand, Ce is a planar curve of degree greater than 3, so Ce cannotsatisfy a nontrivial cubic equation. Therefore we conclude that Q1 (~e) L2 (~e)−Q2 (~e) L1 (~e) is zero on P2

C. Since L1 and L2 cannot be proportional (otherwise

we would be able to obtain from E1 and E2 a quadratic equation in ~e satisfiedby Ce) we conclude by unique factorization that Li is a factor of Qi for i ∈ {1, 2}.Hence Qi (~e) = L (~e) Li (~e) for some linear polynomial L.

From Equation (6) we infer that L1 (~q) ej = L1 (~e) qj for j ∈ {1, 2, 3}. SinceE1 is zero on Cw, we have −L1 (~q) = L (~e) L1 (~e) on Cw. Multiplying by ej thelast equation yields:

−L1 (~e) qj = −L1 (~q) ej = L (~e) L1 (~e) ej ,

15

implying that qj = L (~e) ej holds on Cw for j ∈ {1, 2, 3}.One can verify that the automorphism corresponding to the translation by

a vector ~a = (a1, a2, a3) ∈ R3 acts on the coordinates of PC(1, 1, 1, 2, 2, 2) bysending

(~e, ~q) 7→ (~e, ~q + `~a ~e) ,

where `~a = a1 e1 + a2 e2 + a3 e3. Hence, if L (~e) = α1 e1 + α2 e2 + α3 e3, it isenough to apply to Cw the automorphism corresponding to the translation bythe vector α = (α1, α2, α3) to get that q1 = q2 = q3 = 0 holds on Cw. Thisproves the statement.

We are now ready to prove our main result.

Theorem 3.10. Let Π be a mobile icosapod such that its configuration curve KΠ

is irreducible and satisfies (†). Then KΠ is a line-symmetric motion.

Proof. From the discussion so far (Proposition 3.2, Lemma 3.5 and the para-graph following Lemma 3.7) we know that it is possible to apply a projective au-tomorphism toKΠ so that the equationsmij = mji andm11+m22+m33+h = 0hold. Hence we only need to ensure x = y. However, in the new embed-ding in PC(1, 1, 1, 2, 2, 2) defined in Lemma 3.9 those equations correspond toq1 = q2 = q3 = 0, so Lemma 3.9 shows the claim.

Remark 3.11. From Theorem 3.10 the set of base and platform points of theicosapod possesses a line-symmetry during the complete self-motion. But thisproperty holds for any pod with a line-symmetric self-motion (see Theorem 1.1).As a consequence one could call these mechanical linkages "line-symmetric icos-apods" by analogy to the "line-symmetric Bricard octahedra".

4. Construction of real icosapods

Borel proposed the construction of line-symmetric icosapods simply by inter-secting Xinv with a general linear subspace T of dimension 7 in P10

C, as explained

in Section 4.1. Since Xinv is a variety of codimension 6 and degree 12 in P10C,

the intersection C = Xinv ∩ T is an irreducible curve of degree 12. Indeed, C isa canonical curve of genus 7, as one can read off from the Hilbert series of Xinv.The projection Ce of C to the Euler parameters is a planar sextic. Recall thatthe leg set LC is the intersection of Y with the dual of span(C), a linear sub-space of codimension 7; this is, in general, a set of 20 complex points. It was notpreviously known whether there are examples with 20 real legs, and the goal ofthis section is to show that there are instances of such curves C for which thisis the case. We reduce the problem to a question on spectrahedra whose answeris well-known.

4.1. Borel’s constructionWe rephrase Borel’s construction using the terminology and concepts intro-

duced in this paper. Let S be the linear subspace defined by the equationsM = M t and x = y. Notice that S has projective dimension 10 and con-tains Xinv, although span(Xinv) is a linear subspace of dimension 9. The re-striction of the bilinear sphere condition from Equation (3) can be written in a

16

symmetric way as

l h+ z00r − 2

3∑i=1

s0ixi − 2

3∑i=1

ziimii − 2

3∑1≤i<j

sijmij = 0, (7)

where sij = zij + zji for 1 ≤ i < j ≤ 3. We denote this equation by SBSC, forsymmetric bilinear sphere condition. It defines a duality between S and a linearsubspace P̌10

C⊆ P̌16

Cwhose projective coordinates are l,z00, . . . , z33,s01, . . . , s23.

The intersection of the leg variety Y with such a P̌10C

parametrizes pairs of legsobtained by swapping the roles of the base and the platform points. Denote byπ : P̌10

C99K P̌9

Cthe projection defined by removing the l-coordinate. We denote

by Yinv the image of the map α : P3C× P3

C−→ P̌9

C,(

(a0 : · · · : a3), (b0 : · · · : b3))7→ (a0 b0︸︷︷︸

z00

: · · · : a3 b3︸︷︷︸z33

: a0 b1 + a1 b0︸ ︷︷ ︸s01

: · · · : a2 b3 + a3 b2︸ ︷︷ ︸s23

).

One can easily prove that Yinv is nothing but the projection under π of theintersection Y ∩ P̌10

C. Note that α is a 2 : 1 map, since α(a, b) = α(b, a) for all

a, b ∈ P3C. Because of this, it might happen that two pairs of complex points are

sent by α to a real point of Yinv.For any curve C ⊆ Xinv, the leg set LC is equal to the intersection of

the linear space Γ̃, dual to span(C), with the cone over Yinv in P̌10C, namely

π−1 (Yinv). If dim span(C) = 6, then dim Γ̃ = 3. Since Xinv is contained in thehyperplanem11+m22+m33+h = 0, it follows that Γ̃ passes through the point pewith coordinates l = −2, z11 = z22 = z33 = 1 and all other coordinates beingzero. Borel’s construction can be rephrased as simply choosing a 3-space passingthrough pe and intersecting with the cone over Yinv. This cone has degree 10 andcodimension 3 in P̌10

C, so generically there are 10 solutions (possibly complex),

each corresponding to a pair of legs. For a general 3-space Γ̃ passing throughpe, one can ask three questions on reality:

1. How many of the 10 points of Γ̃ ∩ π−1 (Yinv) are real?

2. How many of the real points above have real preimages under α? Namely,how many real legs does the curve C admit?

3. Does the curve Xinv ∩ Λ have real components, where Λ is the dual to Γ̃under SBSC?

The answers to Question (1) and (2) only depend on the projection of Γ̃ to P̌9C.

In order to obtain positive answers for Question 3, it is also convenient to startwith the projection to P̌9

C.

Definition 4.1. A Borel subspace Γ is a 3-space in P̌9Cpassing through π(pe).

The following proposition settles Question (3).

Proposition 4.2. Let Γ be a Borel subspace. Then there exists a 3-space Γ̃passing through pe such that π(Γ̃) = Γ and Xinv ∩Λ has real components, whereΛ is the dual of Γ̃ under SBSC.

17

Table 1: Points in Γ̃ ∩ π−1(Yinv).

no. of real points 2 4 6 8 10frequency 22 1067 3638 4035 1238

Table 2: Points in α−1(Γ̃ ∩ π−1(Yinv)

).

no. of real points 2 4 6 8 10 12 14 16 18 20frequency 0 4107 0 5240 0 650 0 3 0 0

Proof. Let f : S 99K P4Cbe the projection from the linear subspace U dual

to π−1(Γ). Note that U is contained in the hyperplane m11 +m22 +m33 +h = 0.Hence the image of Xinv under f is contained in a linear 3-space, and f |Xinv

hasone-dimensional fibers. Since Xinv has real components, it follows that thereexist fibers (f |Xinv)−1(q) with real components, for some q ∈ P4

C. We just need

to choose Γ̃ dual to f−1(q); then Xinv ∩ Λ coincides with (f |Xinv)−1(q) andtherefore has real components.

In order to get some statistical data on the answers to Question (1) and (2),we tested 10000 random examples3 of Borel subspaces. The results are shownin Tables 1 and 2.

As one can see, the experimental data do not reveal any example of podswith 20 real legs. This is, however, misleading; see the next section.

4.2. Icosapods via spectrahedraWe conclude our work by showing how it is possible to construct a mobile

icosapod with 20 real legs using some result in convex algebraic geometry.Consider a 4-dimensional vector space A of symmetric 4×4-matrices over R.

Classically, the spectrahedron defined by A is the subset of A comprised ofpositive semidefinite matrices. One can also consider the spectrahedron as asubset of the projective space P(A) ∼= P3. The boundary of the spectrahedronconsists of the semidefinite matrices with determinant 0, and hence its Zariskiclosure is a quartic surface in P3, called the symmetroid defined by A. In general,a symmetroid has 10 double points, corresponding to matrices of rank 2.

Given a spectrahedron whose symmetroid has 10 complex double points, itstype is the pair of integers (a, b), where a is the number of real double pointsof the symmetroid and b is the number of real double points of the symmetroidthat are also contained in the spectrahedron.

Theorem 4.3. There is a bijective correspondence between quartic spectrahedra

containing the matrix E :=

(0 0 0 00 1 0 00 0 1 00 0 0 1

)and Borel subspaces. For a spectrahedron

defined by a vector space A and the corresponding Borel subspace Γ, the followingstatement holds: if the spectrahedron has type (a, b), then Γ intersects Yinv in areal points, and a− b of them have real preimages under α.

3The Maple code used to perform such experiments can be downloaded from http://matteogallet.altervista.org/main/papers/icosapods2015/Icosapods.mpl

18

Proof. We identify P̌9Cwith the projectivization of the vector space of symmetric

4 × 4 matrices in the following way: a point with homogeneous coordinatesz00, . . . , s23 corresponds to the class of the matrix

2 z00 s01 s02 s03

s01 2 z11 s12 s13

s02 s12 2 z22 s23

s03 s13 s23 2 z33

.

A linear subspaceA of dimension 3 in the space of symmetric matrices containingthe matrix E corresponds then to a Borel subspace Γ.

The subvariety Yinv corresponds to the subvariety of matrices of rank 2.A real matrix of rank 2 does not lie on the spectrahedron if and only if thequadratic form defined by it is a product of two distinct real linear forms, andthis is true if and only if its preimage under α is real.

Degtyarev and Itenberg in [48] determined all possible types of quartic spec-trahedra. In particular, spectrahedra of type (10, 0) do exist, hence by Theo-rem 4.3 they provide Borel subspaces intersecting Yinv in 10 real points, each ofthem having two real preimages under the map α. This implies that there existBorel icosapods with 20 real legs.

In [3], the authors give explicit examples of spectrahedra for all possibletypes. The given example of type (10, 0) does not contain the matrix E, but itis easy to adapt their example to one of the same type that does contain E.

4.3. ExampleStarting from [3, Section 2, Case (10, 0)] we computed4 the following exam-

ple, which is suitable for graphical representation:

P1 = p4 =(− 19493

142100 ,−208894325 ,−

249625

), p1 = P4 =

(− 36411

267844 ,−1608

177793 ,504

25399

),

P2 = p5 =(− 269

5000 ,39

1000 ,17500

), p2 = P5 =

(− 47

368 ,−12

1771 ,21

1265

),

P3 = p6 =(− 1863

14645 ,−1068511555400 ,

2509222200

), p3 = P6 =

(− 15185

112462 ,−120

149303 ,48

3047

).

We apply a half-turn to the platform about a line ` through the point(− 1

10 , 0, 0) in direction (1, 700371694410000000000 ,

810 ). In the resulting initial position, which

is illustrated in Figure 1, the squared leg lengths of the first six legs read as fol-lows:

d21 = d2

4 = 1081643179736912972309543483891375692276669953748621822688942197018838171875 ,

d22 = d2

5 = 21948230578108174284480998906129002829339836395492656900000000 ,

d23 = d2

6 = 4185335506762812187908674782558830797636621874987061375644008358435317156000 .

For this input data the self-motion consists of two components. The trajectoriesof the component which passes through the initial position are illustrated inFigures 3 and 4. In the latter figure also the associated basic surface is displayed.An animation of this line-symmetric self-motion can be downloaded from www.geometrie.tuwien.ac.at/nawratil/icosapod.gif.

We close the paper by mentioning two open questions that we find of interest:

4The Maple code containing a similar computation can be downloaded from http://matteogallet.altervista.org/main/papers/icosapods2015/Icosapods.mpl

19

`

Figure 3: The 20 spherical trajectories passing through the initial position of the icosapod.

- Starting from spectrahedra of type (a, b) with a−b ≥ 4 one may constructmobile pods with 16, 12 or 8 legs: is it true that a general mobile podwith 16, 12 or 8 legs is line-symmetric? (It is known that a, b have to beeven numbers.)

- Identify all cases where more than 20 points move on spheres during aline-symmetric motion; i.e. (a) 1-dim, (b) 2-dim or even (c) 3-dim set ofpoints with spherical trajectories. Case (c) is completely known due toBricard [12], but cases (a) and (b) are still open. Examples for both casesare known (cf. Section 1.2.2 and [11, 12, 14, 18]).

Acknowledgments

We would like to thank the anonymous referees for the many useful sugges-tions they provided, which we think helped improving the quality of the paper.We also thank Charles Wampler for pointing us at several references. The firstand third-named authors’ research is supported by the Austrian Science Fund(FWF): W1214-N15/DK9 and P26607 - “Algebraic Methods in Kinematics: Mo-tion Factorisation and Bond Theory”. The second-named author’s research isfunded by the Austrian Science Fund (FWF): P24927-N25 - “Stewart-Goughplatforms with self-motions”.

[1] N. White, W. Whiteley, The algebraic geometry of motions of bar-and-body frameworks, SIAM J. Algebraic Discrete Methods 8 (1) (1987) 1–32.

20

`

Figure 4: The same set-up as in Figure 3 but from another perspective. In addition a strip ofthe base surface of this line-symmetric self-motion is illustrated, where ` is highlighted. Notethat one of the two displayed families of curves on the surface is composed of straight lines,i.e. the generators of the ruled surface.

doi:10.1137/0608001.URL http://dx.doi.org/10.1137/0608001

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